Digital Fresnel holography beyond the Shannon limits

Mathieu Leclercq; Pascal Picart

doi:10.1364/OE.20.018303

1. Introduction

Digital holography found its origin in the early 1960-70’s and was really demonstrated recently [1–3]. The theoretical background of digital holography was presented in previous works, which discussed theory and reconstruction algorithms for digital holography, according to the different possible schemes for the recording, i.e. in-line holography [4,5], off-axis Fresnel holography [6–8], digital Fourier holography [9], and reconstruction with Fresnelets in off-axis holography [10]. Several authors were interested in investigating the image-to-object relationships and the Shannon conditions for recording [9,11–18]. The Shannon conditions are particularly related to the recording sensor, which is described by its spatial extent, pixel pitch and surface. The effect of the finite extension of pixels was considered by several authors. Kreis, and also Guo [7,8,19], proposed a full analytical modeling including the sampling and pixel dimension effect. They concluded that the pixel acts as a multiplicative sinc function in the Fourier space of the hologram. Stern and Javidi [20] described the influence of the finite size of pixel and concluded that the pixel is a convolution smearing factor of the reconstructed field. Furthermore, the modulation transfer function of the pixel is a sinc function. In 2008, Picart and Leval [12] proposed a general theoretical formulation for imaging under Shannon conditions, in which the reconstructed image is considered as a convolution between the real object and the impulse response of the full process. The impulse response includes the pixel function as a convoluting argument. Recently, Kelly et al discussed the practical resolution limits in digital holography [13–15]. They especially concluded that the averaging introduced by the finite size of the pixels degrades the quality of the reconstructed hologram by convolving with a narrow rectangular function that is the same size as the pixel. In their paper, Xu et al proposed an imaging analysis of digital holography, in which they demonstrated that the pixel has a multiplicative sinc effect on the reconstructed field [21]. In 2009, Demoli et al [22] proposed experimental results exhibiting the effect of under sampling in digital Fourier holography. They showed the appearance of aliasing frequencies. Furthermore, their experimental results include the sinc effect of the pixel dimension, although this point was not further discussed in the paper. Additionally, their theoretical modeling neglected the extended pixel dimensions. As it will be seen later in this paper, the image movement they discussed is not in agreement with the results presented here.

In order to clarify the different contradictory approaches, this paper proposes an investigation of the Shannon limits related to the pixel dimension influence. The paper is organized as follows: Section 2 presents the theoretical basics of the recording and reconstruction process in digital holography, the influence of the active surface of pixels and the pixel paradox; Section 3 focuses on the experimental methodology and Section 4 presents the experimental results. Section 5 draws the conclusions of the study.

2. Theoretical basics

2.1 Digital hologram recording and reconstruction

Let us consider the imaging of digital holograms in a smooth-plane reference wave scheme. In the recording plane, where an object wave interferes with a reference wave, the hologram is expressed as:

H = {| R |}^{2} + {| O |}^{2} + R_{}^{*} O + R O^{*},

where R(x,y) = a_Rexp[2iπ(u₀x + v₀y)] is the reference wave with spatial frequencies {u₀,v₀}, and O is the wave diffracted in the recording plane by the object located at distance d₀ from this plane. In the Fresnel approximations, we have [23]:

\begin{matrix} O (x, y, d_{0}) = - \frac{i}{λ d_{0}} \exp (\frac{2 i π d_{0}}{λ}) exp (\frac{i π}{λ d_{0}} (x^{2} + y^{2})) \\ \times \int \int A (X, Y) exp (\frac{i π}{λ d_{0}} (X^{2} + Y^{2})) exp (- \frac{2 i π}{λ d_{0}} (x X + y Y)) d X d Y, \end{matrix}

and A(X,Y) = A₀(X,Y)exp[iψ₀(X,Y)] is the object wave front. Due to the spatial extension of the pixel, the detected holograms, which will be transferred to the digital acquisition board, are composed of elementary spatially-integrated zones of the real ones. At any pixel coordinate (lp_x,kp_y) at which the digital hologram is recorded (l, k: integers; p_x, p_y: pixel pitches), we have:

H_{D} (l p_{x}, k p_{y}, d_{0}) = \int_{l p_{x} - Δ_{x} / 2}^{l p_{x} + Δ_{x} / 2} \int_{k p_{y} - Δ_{y} / 2}^{k p_{y} + Δ_{y} / 2} H (x, y, d_{0}) d x d y,

Introducing the pixel function defined as an even “rect” function,

rect (\frac{x}{Δ_{x}}, \frac{y}{Δ_{y}}) = {\begin{cases} 1 if | x | \leq Δ_{x} / 2 and | y | \leq Δ_{y} / 2 \\ 0 if not \end{cases},

Equation (3) is rewritten according to the following convolution formula [12]:

H_{D} {(x, y, d_{0})}_{x = l p_{x}, y = k p_{y}} = H (x, y, d_{0}) * rect (\frac{x}{Δ_{x}}, \frac{y}{Δ_{y}}) .

In Eq. (5), * means two-dimensional convolution. Note that the above-mentioned papers describe the pixel effect by almost the same parameter: a narrow rectangular function as given in Eq. (4).

Considering the discrete recording, the reconstruction of the object field at distance −d₀ from the recording plane is given by the discrete Fresnel transform [3,6]:

\begin{array}{l} A_{r} (X, Y, - d_{0}) = - \frac{i \exp (2 i π d_{0} / λ)}{λ d_{0}^{}} \exp [- \frac{i π}{λ d_{0}} (X^{2} + Y^{2})] \\ \times \sum_{k = 0}^{k = K - 1} \sum_{l = 0}^{l = L - 1} H_{D} (l p_{x}, k p_{y}, d_{0}) \exp [- \frac{i π}{λ d_{0}} (l^{2} p_{x}^{2} + k^{2} p_{y}^{2})] \exp [\frac{2 i π}{λ d_{0}} (l X p_{x} + k Y p_{y})] . \end{array}

2.2 The pixel paradox

Several authors discussed the influence of the pixel function on the reconstructed object. According to [12], the pixel induces a spatial low-pass filtering of the reconstructed object. This means that, mathematically, the reconstructed object is a convolution between the real object and the pixel function. This point was thoroughly investigated in references [13–15,20]. This result is quite compatible with optical imaging properties. Indeed, in classical imaging, where the object is projected onto the sensor plane by means of an optical lens, the image is the real one convoluted by the point spread function of the lens and convoluted by the pixel function (see Eq. (5)). So, in classical imaging, the filtering of the real object is due to the lens and the pixel dimensions. Digital holography is an interferometric method to image objects. In such non-conventional imaging, filtering is not processed by a lens since there is no lens in the set-up. However, it is performed by the double free-space propagation, due, first, to the physical diffraction from object to sensor, and, second, to the numerical diffraction from sensor plane to the reconstructed plane. In addition, the results obtained with digital image-plane holography are quite similar to those obtained with digital Fresnel holography [24]. Finally, the fundamental results given in references [12–15,20] are common sense and can be intuitively understood.

Although it may seem strange regarding this analysis, previously published papers [9,21] argued that the pixel function induces a multiplicative sinc function to the reconstructed object amplitude. In order to explain this point of view as straightforwardly as possible, let us consider that the object amplitude in the recorded plane is slowly varying compared to the pixel size. As a result, Eq. (3) can be re-written as:

\begin{matrix} H_{D} (l p_{x}, k p_{y}, d_{0}) \propto {| O (l p_{x}, k p_{y}, d_{0}) |}^{2} + {| R (l p_{x}, k p_{y}) |}^{2} \\ + O (l p_{x}, k p_{y}, d_{0}) R^{*} (l p_{x}, k p_{y}) \sin c (π u_{0} Δ_{x}) \sin c (π v_{0} Δ_{y}) \\ + O^{*} (l p_{x}, k p_{y}, d_{0}) R (l p_{x}, k p_{y}) \sin c (π u_{0} Δ_{x}) \sin c (π v_{0} Δ_{y}) . \end{matrix}

In Eq. (7), sinc(x) = sin(x)/x. According to Eq. (7), the useful + 1 order is amplitude-modulated by the sinc functions; therefore, the reconstructed object will also be amplitude-modulated by the sinc functions [21].

These two approaches stand for what can be called the pixel paradox. Indeed, the first approach states that the pixel function is a convolution function, whereas the latter states that it is a multiplicative sinc function. One may wonder what magical reason explains why the same element provides different influences. Furthermore, note that the Fourier transform of the recorded hologram provides the spatial frequency spectrum of the object. Thus, it is multiplied by the sinc function according to Eq. (5). Nevertheless, the Fourier transform of the hologram is also the discrete Fresnel transform for a reconstruction distance tending toward infinity, leading to an out-of-focus object. There is no logical reason for the out-of-focus object to be not convoluted by the pixel function in the same manner as the focused object will be. But it is quite common sense that the Fourier transform of the hologram is multiplied by the sinc function.

2.3 The Shannon limits

The Shannon limits correspond to the recording conditions compatible with the Shannon theorem. There are several ways to optimize the recording condition, so that the three diffraction orders may not overlap [12,16,25]. This paper deals with an off-axis configuration; the optimization proposed in [12] is considered. The Shannon theorem applied to off-axis digital holography, resulting in the spatial separation of the three diffraction orders, leads to the optimal recording distance. Practically, the spatial frequencies can be adjusted following this method: the reference beam is perpendicular to the recording plane but the object is laterally shifted [22,24]. We consider here that the spatial Shannon limits are given by the maximum lateral shift, as long as that the center of the object fulfills the Shannon theorem. When translating the object along the horizontal axis of a quantity ΔX, the mean spatial frequency of the hologram becomes u₀ = ΔX/λd₀. When laterally moving the object in the field of view, the reconstructed object will be localized at the corresponding spatial shift until spatial aliasing occurs. Indeed, the field of view reconstructed by the discrete Fresnel transform is sized λd₀/p_x × λd₀/p_y. Because of the sampling of recording and reconstruction spaces, the field of view is virtually periodic. This means that the reconstructed field is repeated infinitely, with a period equal to its width. However, the numerical computation can only give the data of the main window corresponding to the field of view. Figure 1 illustrates the virtual periodicity of the reconstructed field. The central window (not translucent) is that reconstructed by the FFT algorithms, whereas the 8 other ones (translucent) are not reachable by FFT calculation. However, these 8 neighborhood zones indicate that, when the object is shifted from the central window to one of the adjacent ones, its reconstructed image comes from the symmetric zone. Figure 1(left) shows the aliasing that occurs when exceeding Shannon limits: the under-sampled part of the object is shifted out of the main window and re-appears in the opposite corner. Thus, for a physical object shifted along the x-axis with ΔX∈{(2k−1)λd₀/2p_x,(2k + 1)λd₀/2p_x}, k integer, the position of the numerically-reconstructed object is given by ΔX_r = ΔX−kλd₀/p_x. Beyond the Shannon limits, the reconstructed object appears in the field of view at cyclic positions and with an amplitude attenuated by the sinc(πΔXΔ_x/λd₀) function.

Fig. 1 Virtual periodicity of the reconstructed field and lateral shift of the object.

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As an illustration, Fig. 2 shows the Shannon limits given by the numerical sampling and the amplitude modulation of the reconstructed object according to both the mean spatial frequency of the digital hologram and the lateral shift of the object. In this example, the filling factor of the sensor is taken to equal to 1 (i.e. Δ_x = p_x).

Fig. 2 Shannon limits and amplitude modulation vs lateral shift of the object.

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The next sections present experiments that establish the sinc influence of the sensor pixel and the cyclic position beyond the Shannon limits.

3. Experimental set-up and methodology

The experimental set-up is described in Fig. 3 . It is based on an off-axis Mach Zehnder configuration in which the sensor has 8 bits digitization with L × K = 1024 × 1360 pixels sized p_x = p_y = 4.65μm. The laser is a continuous frequency doubled NdYAG laser (λ = 532nm, P₀ = 150mW) and the object is a 50 cents euro coin, 24mm in diameter. It is illuminated with a circular spot. The object is mounted on a translation stage (Fig. 3); the mean horizontal spatial frequencies of the digital hologram, {u₀}, may thus vary continuously. The amplitude modulation due to the pixel is then formulated with sinc(πΔ_xΔX/λd₀). In order to measure the amplitude modulation of the object, a power meter estimates the injected laser power into the interferometer. A rotation stage (mirror M2) ensures that the object is correctly illuminated when translating in the x-axis direction. Note that change in the illuminating angle, due to the object translation, requires the measurement of the bidirectional reflectance distribution function (BRDF) of the object for angles similar to those provided by the illumination.

Fig. 3 Optical setup for investigating the pixel influence.

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The BRDF was taken into account for the amplitude modulation of the reconstructed object so that it may only be due to the pixel effect and not to photometric considerations (an experiment not described in this paper). The object is localized at a constant distance d₀ = 900mm during the experiment. The object can be slightly tilted on its mount. This facility provides the possibility to study the effect of the pixel dimension on phase differences extracted between two object states, especially the decorrelation noise and the coherence degree of the two speckle phases [24].

The methodology is the following: starting from ΔX = 0 (object localized at the upper center of the reconstructed field (main window), we record two digital holograms for each lateral shift of the object (shift oriented from right to left). A small tilt is applied to the object between the two recordings. At the same moment, the laser power is measured in order to correct fluctuations. Then, the digital holograms are computed for each shift. The amplitude modulation and the cyclic position of the reconstructed image can thus be evidenced. The amplitude modulation is calculated by averaging the reconstructed amplitude over a set of 50 × 50 pixels, then normalized with the amplitude at the 0 shift. The reconstructed phases are also extracted and for each object shift and the phase change is calculated modulo 2π. The tilt of the object is adjusted so that it may produce only few fringes, to limit the speckle decorrelation. Thus, the study of the influence of the noise induced by the out-of-Shannon recording can be investigated for the phase change measurement. Noise can be measured according to [26]. The subtraction of the low-pass filtered phase difference from the raw phase difference leads to an estimation of the standard deviation of the noise included in the raw data. The reconstructed object phase is random and has the properties of a speckle phase, since it is closely related to the object rough surface. So, correlation properties are related to the second-order probability density function of the phase [27, p. 406]. The analytical calculation of the joint probability density function of the phases ψ₁ and ψ₂ of two speckle patterns is a difficult one and will not be detailed in this paper. The reader is invited to look at references [27, p. 406] and [28, p. 163]. We note ε = ψ₁−ψ₂ the noise induced both by the speckle decorrelation and amplitude modulation between two object fields reconstructed after tilting the object, and we note Δφ the phase change due to the object tilt. Then ψ₂ = ψ₁ + ε + Δφ, Δφ being considered as a deterministic variable. The probability density function of ε depends on the modulus of the complex coherence factor |μ| between the two speckle fields. With β = |μ|cos(ε), the second-order probability density of the phase noise ε is given by:

p (ε) = \frac{1 - | μ |}{2 π} {(1 - β^{2})}^{- 3 / 2} (β \sin^{- 1} β + \frac{π β}{2} + \sqrt{1 - β^{2}}) .

Equation (8) can be used as a relevant indicator so as to compare the noise sensitivity at different lateral shifts, the correlation factor |μ| being a quality marker extracted from experimental data.

4 Experimental results

The object is shifted along the X-axis direction from 0 to 280mm. The Shannon limits for the set-up is 51.48mm for the lateral position. The BRDF for the experimental conditions was measured and was transposed as a lateral shift variation. It was evidenced that its contribution to the amplitude modulation is a multiplication by exp(−ΔX/b). This means that the amplitude modulation on the reconstructed object is a f(ΔX) = |sinc(πΔX/a)exp(−ΔX/b)| function rather than |sinc(πΔX/a)|. Figure 4 shows the measured amplitude modulation from 0 to 280mm. The experimental curve is compared to the analytical model f(ΔX), giving a = 119.15mm and b = 168.35mm (99.824% correlation). The |sinc(πΔX/a)| curve is also plotted. Fit f(ΔX) is in excellent agreement with the experimental curve. This confirms a part of the pixel paradox discussed in this paper: the pixel dimension has a sinc influence on the reconstructed object amplitude. Furthermore, this validates the modeling for the BRDF of the object, that must be taken into account to measure pixel influence.

Fig. 4 Amplitude modulation vs lateral object shift.

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Figure 5 shows the reconstructed fields when the object is shifted (zero order removed for better visibility), corresponding to red points 1 to 6 in Fig. 4. Point 1 corresponds to the initial 0-shift position, point 2 to the Shannon limit, point 4 to the first zero-crossing of the sinc function and points 5-6 to its second lobe. The object always moves in the same direction, which is from right to left. These results are in good agreement with the previous discussion about Fig. 1 and theoretical basics. Note that the results are in contradiction with [22], in which the object direction may be inverted.

Fig. 5 Experimental visualization of the cyclic periodicity and sinc attenuation (Media 1)

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For each object shift, the phase changes are extracted from the two recordings without and with tilt. Then, the noise and the probability density are estimated; Eq. (8) is fitted to the experimental data; the noise rms and the correlation factor are calculated. Figure 6 shows the raw and filtered phases (5 × 5 moving average according to [29]) extracted from measurements. Tilt was applied to the object with an almost good reproducibility. This acceptable reproducibility can be appreciated on experimental results in Fig. 6, although the tilt angles are not exactly identical. However, this reproducibility is quite sufficient to maintain the speckle decorrelation as constant, so that the variation of noise may be attributed to out-of-Shannon conditions. Figure 6(a) and 6(b) show the 6 phase changes corresponding to 6 red points (1 to 6 in Fig. 4).

Fig. 6 Phase changes extracted from the tilt measurement, (a) raw, (b) filtered

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Figure 6 shows that the fringe contrast is relatively good for recordings beyond the Shannon limits. The fringes obtained for points 1 and 2 do not seem to be altered by the sinc. The fringe maps obtained for point 3 (near zero-crossing) and point 5 (second lobe) reveal good quality. This qualitative analysis is confirmed by a quantitative measurement of the noise rms and speckle correlation factor [24,26]. Figure 7 shows the noise rms and the correlation factor |μ| of Eq. (8) according to the object lateral shift. The Shannon limit and the red points of Fig. 4 are also shown.

Fig. 7 Noise rms and correlation factor vs the lateral shift of the object

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The two curves are well correlated to the sinc variation. The zero-crossings of sinc induce an increase in noise (two visible peaks) and a decrease of the correlation between the two phases (two visible craters). Figure 7 shows that the noise rms and the correlation factor remain constant beyond the Shannon limit, until a shift of about ΔX≈90mm, corresponding to 1.74 times the Shannon limit.

The filling factor of the CCD sensor can be estimated thanks to approximating the amplitude modulation of Fig. 1. The first zero-crossing of the sinc function is obtained for a = 119.15mm. Thus, the pixel width is estimated to Δ_x = λd₀/a = 4.01μm, leading to a filling factor of Δx/p_x = 86.4%.

5. Conclusion

This paper has experimentally demonstrated the influence of the pixel dimension in digitally-reconstructed holograms. Extended pixels induce a sinc-type amplitude modulation which depends on the mean spatial frequency of the hologram. As a practical consequence of the experimental results, the filling factor of the sensor can be estimated. The demonstration of the sinc modulation contributes to solving a part of the pixel paradox. The paper has demonstrated that recording beyond the Shannon limits is possible since both amplitude and phase information can be retrieved. Furthermore, the phase changes of the object can be obtained with a very good contrast and may be only limited by the decorrelation noise, as when the Shannon conditions are fulfilled. As a result, classical Shannon constraints can be relaxed without deteriorating the quality of the image reconstruction.

Acknowledgments

This research is funded from the French National Agency for Research (ANR) under grant agreement n°ANR 2010 BLAN 0302.

References and Links

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Digital Fresnel holography beyond the Shannon limits

Abstract

1. Introduction

2. Theoretical basics

2.1 Digital hologram recording and reconstruction

2.2 The pixel paradox

2.3 The Shannon limits

3. Experimental set-up and methodology

4 Experimental results

5. Conclusion

Acknowledgments

References and Links

Supplementary Material (1)

Cited By

Figures (7)

Equations (8)

Optics Express